summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=ln (x + 5)-4 )\na. the function ( f ) is decreasing on the subinterval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never decreasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local maximum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local maximum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local minimum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local minimum.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=ln (x + 5)-4 )\na. the function ( f ) is decreasing on the subinterval(s)\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never decreasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local maximum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local maximum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local minimum at ( x = )\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no local minimum.

Answer

Explanation:

Step1: Find the domain of the function

The domain of (y = \ln(x + 5)-4) is determined by the argument of the natural logarithm. For (y=\ln(u)), (u>0). Here (u=x + 5), so (x+5>0) or (x>- 5). The domain is ((-5,\infty)).

Step2: Find the first - derivative

Using the chain rule, if (y=\ln(x + 5)-4), then (y^\prime=\frac{d}{dx}(\ln(x + 5))-\frac{d}{dx}(4)). Since (\frac{d}{dx}(\ln(u))=\frac{u^\prime}{u}) (where (u=x + 5) and (u^\prime = 1)) and (\frac{d}{dx}(4)=0), we have (y^\prime=\frac{1}{x + 5}).

Step3: Analyze the sign of the first - derivative

For (x\in(-5,\infty)), (x + 5>0), so (y^\prime=\frac{1}{x + 5}>0) for all (x\in(-5,\infty)). Since the first - derivative (y^\prime>0) for all (x) in the domain of (y = f(x)), the function is increasing on ((-5,\infty)) and never decreasing. For a local maximum or minimum, we look at the critical points. Critical points occur where (y^\prime = 0) or (y^\prime) is undefined. (y^\prime=\frac{1}{x + 5}) is never equal to (0) (because (1\neq0) for any (x)) and is undefined at (x=-5) (but (x =-5) is not in the domain of (y = f(x))).

Answer:

B. The function (f) is never decreasing. B. The function (f) has no local maximum. B. The function (f) has no local minimum.